## 2.1: Derivatives and Rates of Change

Tangents

1. Definition: The tangent line to the curve $y=f(x)$ at the point $\displaystyle P(a, f(a))$ is the line through $\displaystyle P$ with slope

$\displaystyle m=\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$

provided that this limit exists

– there is another expression for the slope of a tangent line that is sometimes easier to use. If $\displaystyle h=x-a$, then $\displaystyle x=a+h$ and so the slope:

$\displaystyle m=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$

– Velocities

we define the velocity (or instantaneous velocity$\displaystyle v(a)$ at time $\displaystyle t=a$ to be the limit of these average velocities:

$\displaystyle v(a)=\lim_{h \to0}\frac{f(a+h)-f(a)}{h}$

– Derivatives

Definition: The derivative of a function $\displaystyle f$ at a number $\displaystyle a$, denoted by $\displaystyle {f}'(a)$, is

$\displaystyle {f}'(a)= \lim_{h \to0 }\frac{f(a+h)-f(a)}{h}$

if this limit exists

– if we write $\displaystyle x=a+h$, then we have $\displaystyle h=x-a$ and $\displaystyle h$ approaches $0$ if and only if $\displaystyle x$ approaches $\displaystyle a$. therefore an equivalent way of stating the definition of the derivative, as we saw in finding tangent ines, is

$\displaystyle {f}'(a)=\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$

– The tangent line to $\displaystyle y=f(x)$ at $\displaystyle (a, f(a))$ is the line through $\displaystyle (a, f(a))$ whose slope is equal to $\displaystyle {f}'(a)$, the derivative of $\displaystyle f$ at $\displaystyle a$

– Rates of Change

instantaneous rate of change = $\displaystyle \lim_{\Delta x \to0 }\frac{\Delta y}{\Delta x}=\lim_{x_{2} \to x_{1}}\frac{f(x_{2})-f(x_{1})}{x_{2}- x_{1}}$

– The derivative $\displaystyle {f}'(a)$ is the instantaneous rate of change of $\displaystyle y=f(x)$ with respect to $\displaystyle x$ when $\displaystyle x=a$

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